Post 1 — Qubits and the Bloch Sphere

This first entry introduces the Bloch sphere representation of qubits and demonstrates how to obtain Bloch vectors with the accompanying Python code.

A pure qubit state is written as \[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \quad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1. \]

From this vector we form the density matrix \[ \rho = |\psi\rangle \langle \psi |. \]

The Pauli matrices are

\[ \sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. \]

The Bloch vector is defined by \[ r_i = \operatorname{Tr}(\rho \, \sigma_i), \quad i \in \{x,y,z\}. \]

For pure states we obtain \(\|r\|=1\), so every qubit corresponds to a point on the unit sphere \(S^2\).

The function state_to_bloch implements this mapping. Example:

import numpy as np
import hottbloch as h
 
# Basis states
ket0 = np.array([1, 0], dtype=complex)
ket1 = np.array([0, 1], dtype=complex)
 
print("Bloch vector of |0>:", h.state_to_bloch(ket0))
print("Bloch vector of |1>:", h.state_to_bloch(ket1))
 
# Superposition state |+> = (|0> + |1>)/√2
ket_plus = (ket0 + ket1) / np.sqrt(2)
print("Bloch vector of |+>:", h.state_to_bloch(ket_plus))

Expected output:

$|0\rangle$ maps to the north pole $(0,0,1)$.

$|1\rangle$ maps to the south pole $(0,0,-1)$.

$|+\rangle$ maps to a point on the equator $(1,0,0)$.

To plot Bloch points or loops, use:

python hottbloch.py --out ./hott_outputs --loop equator --theta 0

This produces a static Bloch sphere with a marked state. Images are saved in `./hott_outputs`.

This establishes the basic correspondence between qubit states and points on the Bloch sphere. In subsequent posts we extend this to continuous paths, closed loops, and their associated geometric phases.